Raoult term

Curves resulting from Eq. (4.231) are named Kohler curves (Fig. 4.13). The first term on the right side of the Kohler equation is called the Kelvin term and the second is called the Raoult term. From Eq. (4.231) follows the droplet radius for 6 = / // = 1 ... 

Fig. 3 Kohler curve, showing the contributions of the Kelvin and Raoult terms. Reproduced with permission from [299]...

The laboratory studies discussed in this section highlight the importance of the solubility of aerosol organic material oti CCN activity. As stated previously, solubility affects the Raoult term in the Kohler equation (1). The solubility limits the amount of organic material that is incorporated into the particle and thus influences the potential for the formation of a surface film and/or micelles. Hence, solubility indirectly also impacts the surface tension of the particle, thereby influencing the CCN activity via the Kelvin effect. 

At the end of the 1930s, the only generally available method for determining mean MWs of polymers was by chemical analysis of the concentration of chain end-groups this was not very accurate and not applicable to all polymers. The difficulty of applying well tried physical chemical methods to this problem has been well put in a reminiscence of early days in polymer science by Stockmayer and Zimm (1984). The determination of MWs of a solute in dilute solution depends on the ideal, Raoult s Law term (which diminishes as the reciprocal of the MW), but to eliminate the non-ideal terms which can be substantial for polymers and which are independent of MW, one has to go to ever lower concentrations, and eventually one runs out of measurement accuracy . The methods which were introduced in the 1940s and 1950s are analysed in Chapter 11 of Morawetz s book. 

Raoult s law over the whole range of composition, this is because the total change in the free energy, on mixing the particles, is given by nAkT In xA + n0kT In xB, where xA is the mole fraction of A in the solution and xB is the mole fraction of B. For a solution that contains in addition nc particles of species C, and so on, terms must be added, thus... 

Raoult expressed his results in terms of the molecular depression , for a mol. of solute in 100 grams of solvent. The volume of the solvent is 100/p, and this may be taken as the volume of the dilute solution. The corresponding osmotic pressure P, on the assumption that the law of proportionality holds good at this concentration (which is only a fictitious extrapolation) is given by ... 

The behaviour of most metallurgically important solutions could be described by certain simple laws. These laws and several other pertinent aspects of solution behaviour are described in this section. The laws of Raoult, Henry and Sievert are presented first. Next, certain parameters such as activity, activity coefficient, chemical potential, and relative partial and integral molar free energies, which are essential for thermodynamic detailing of solution behaviour, are defined. This is followed by a discussion on the Gibbs-Duhem equation and ideal and nonideal solutions. The special case of nonideal solutions, termed as a regular solution, is then presented wherein the concept of excess thermodynamic functions has been used. 

These relationships are alternative expressions of Raoult s and Henry s laws. Raoulhan and Henrian behaviours of a component in a solution, in terms of activity, are shown schematically in Figure 3.7. 

The standard term p is the chemical potential of the pure component i (i.e. when Xj = 1) at the temperature of the system and the corresponding saturated vapour pressure. According to the Raoult law, in an ideal mixture the partial pressure of each component above the liquid is proportional to its mole fraction in the liquid,... 

Example 2.7. To show what form the energy equation takes for a two-phase system, consider the CSTR process shown in Fig. 2.6. Both a liquid product stream f and a vapor product stream F (volumetric flow) are withdrawn from the vessel. The pressure in the reactor is P. Vapor and liquid volumes are and V. The density and temperature of the vapor phase are and L. The mole fraction of A in the vapor is y. If the phases are in thermal equilibrium, the vapor and liquid temperatures are equal (T = T ). If the phases are in phase equilihrium, the liquid and vapor compositions are related by Raoult s law, a relative volatility relationship or some other vapor-liquid equilibrium relationship (see Sec. 2.2.6). The enthalpy of the vapor phase H (Btu/lb or cal/g) is a function of composition y, temperature T , and pressure P. Neglecting kinetic-energy and potential-energy terms and the work term,... 

Historically, an ideal solution was defined in terms of a liquid-vapor or solid-vapor equilibrium in which each component in the vapor phase obeys Raoult s law. 

Table VII collects the results for all monovalent ion systems for which spectroscopic data are available. Studies of preferential solvation are still at a stage comparable to the establishment of Raoult s and Henry s laws for binary nonelectrolyte solutions. Correlation with thermodynamic data is encouraging for isodielectric solvent systems, but further consideration of the electrostatic terms necessary in the discussion of other systems is required. It is hoped that this present work, which coordinates, correlates, and advances progress made by other workers (7, 18,19, 20, 45, 46, 61, 62, 66, 67, 68), will stimulate systematic experimental investigations of suitable systems by both spectroscopic and thermodynamic methods.

Raoult s and Henry s law are identical, that is, the HR term is the vapor pressure of the pure component, p°. 

A plot of 2 vs. -t2 for symmetrical systems (i.e., ii vo) is shown in Fig. 1 for a series of values of the heat lerm, It shows how the partial vapor pressure of a component of a binary solution deviates positively from Raoult s law more and mure as the components become more unlike in their molecular attractive forces. Second, the place of T in die equation shows that tlic deviation is less die higher the temperature. Third, when the heat term becomes sufficiently large, there are three values of U2 for the same value of ay. This is like the three roots of the van der Waals equation, and corresponds to two liquid phases in equilibrium with each other. The criterion is diat at the critical point the first and second partial differentials of a-i and a are all zero. 

Equilibrium vapor pressure is the vapor pressure of a system in which two or more phases or a substance coexist in equilibrium. In meteorology, the reference is to water substance, unless otherwise specified, If the system consists of moist air in equilibrium with a plane surface of pure water or ice, the more specialized term saturation vapor pressure is usually employed, in which case, the vapor pressure is a function of temperature only. In the atmosphere, the system is complicated by the presence of impurities in liquid or solid water substance (see also Raoult s Law), drops or ice crystals or both, existing as aerosols and, in general, the problem becomes one of nucleation. For example, the difference in vapor pressure over supercooled water... 

In more fundamental terms, the solubility of a chemical in water is determined by the activity coefficient in water yw which can be viewed as a "correction factor" to Raoults Law, i.e.,... 

The relation between Raoult s law and the definition of an ideal solution given by Equation (8.57) is obtained by a study of Equation (10.35) or (10.40). If a solution is ideal, then A/i must be zero and the right-hand side of both equations must be zero. If we write Pyt in both equations as Pt, the partial pressure of the component, and Pyj in Equation (10.40) as P[, then the logarithmic term becomes lnfP P ), which is zero when Raoult s law, given in the form Pl = P[xl, is obeyed. We then see that to define an ideal solution in terms of Raoult s law and still be consistent with Equation (10.57) requires that the experimental measurements be made at the same total pressure and that the vapor behaves as an ideal gas. 

Because deviations from Raoult s law are presumably small, 8 on the right side may be replaced by its Raoult s-law value. For the two terms,... 

Clearly, when B22 = B, the term in square brackets equals 1, and the pressure deviation from the Raoult s-law value has the sign of fin this is normally negative. When the virial coefficients are not equal, a reasonable assumption is that species 2, taken here as the heavier species (the one with the smaller vapor pressure) has the more negative second virial coefficient. This has the effect of making the quantity in parentheses negative and the quantity in square brackets < 1. However, if this latter quantity remains positive (the most likely case), the sign of fin still determines the sign of the deviations. 

Px relation of Raoult s law, and the system therefore exhibits negative deviations. When the deviations become sufficiently large relative to the difference between the two pure-species vapor pressures, the Px curve exhibits a minimum, as illustrated in Fig. 12.96 for the chloroform/tetrahydrofuran system at 30°C. This figure shows that the Py curve also has a minimum at the same point. Thus at this point where x - y the dew-point and bubble-point curves are tangent to the same horizontal line. A boiling liquid of this composition produces a vapor of exactly the same composition, and the liquid therefore does not change in composition as it evaporates. No separation of such a constant-boiling solution is possible by distillation. The term azeotrope is used to describe this state. 

---